7 December 2010
Firm Efficiency, Industry Performance and the Economy: Three-Way Decomposition with an Application to Andalusia
Antonio F. Amoresa,b* and Thijs ten Raab
a Pablo de Olavide University
Department of Economics, Quantitative Methods and Economics History
Ctra. Utrera Km. 1, 41013 Sevilla, Spain
Phone +34 9549 77980. Fax +34 9543 49339
E-mail:
b Tilburg University
Faculty of Economics and Business
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Phone +31 (0)13 466 2365 Fax +31 (0)20 420 6502
E-mail:
*Corresponding author
Abstract.
An economy may perform better because the firms become more efficient, the industries are better organized, or the allocation between industries is improved. In this paper we extend the literature on the measurement of industry efficiency (a decomposition in firm contributions and an organizational effect) to a third level, namely that of the economy. The huge task of interrelating the performance of an economy to industrial firm data is accomplished for Andalusia.
Keywords: Input-Output, industrial organization, comparative advantage, allocative efficiency, efficiency decomposition.
1. Introduction
Inefficiencies abound at the micro, meso and macro level of the economy. Firms do not apply best-practices; industries may be organized suboptimally—with too many or too few firms—and the resources of the economy may be misallocated between industries. These concerns are the subject of the theory of the firm, industrial organization, and macro-economics, but are rarely connected. There are two reasons of this shortcoming. First, in the theoretical literature the focus of efficiency analysis is on the aggregation issue. Two levels are distinguished and there are more gains to be made than at the lower level: gains to trade in a system of regions or gains to reorganization in an industry. In this paper we extend the analysis to more levels. Second, modern economies comprise many industries and very many firms and it is a daunting task to express their performance in terms of the micro data. This paper makes a first attempt.
In the next section, we review a measure for the industrial organization efficiency. In section 3, we propose an inclusion of the industrial specialization efficiency in the economy. In section 4, the economy-wide efficiency is analyzed and decomposed. An application is presented in section 5. The paper ends with some conclusions. Three appendices with a demonstration and data details and procedures are provided, along with a supplementary spreadsheet file containing detailed results.
2. Review of Organization Efficiency
This approach is based on the efficiency gains from a reallocation of resources between firms.
Denote the input and output vectors of firm i in industry k by and , , where Ik is the set of the firms of the industry k and K is the set of the industries in the economy. e is a unitary vector of suitable dimension. The firm efficiency, , is the solution to the following linear program:
(1)
The Because a feasible solution to (1) is a reproduction of firm ik (by putting λik = 1 and all other weights 0) the efficiency score ranges between 0 and 1. This is a Data Envelopment Analysis (DEA) model[1] with Constant Returns to Scale and Output orientation (DEA CRS-O) and inclusion of constant eTyik (which is total production of firm i of industry k, T is transposition) in the objective function; this monotonic transformation will prove useful for the price normalization.
The approach consists in the calculation of the DEA CRS-O score for each firm, using as reference set its industry. The dual program is:
(2)
Here and are the dual variables, solve each program and match the shadow prices of the constraints of (1). By the main theorem of linear programming, the primal and the dual programs have equal solution values: .
The efficiency of industry k, , is the solution to the next program:
(3)
where is again a number between 0 and 1. Superscript k denotes the component k of vector y, the primary output of industry k. Superscripts -k denote the other components, the secondary outputs.[2] The idea is to reallocate the industry inputs, as to maximize k-specific output, inflating it by the expansion factor . Non-specific aggregate output, , may also be expanded, but not necessarily in the same proportion. Since it remains at least the same, our expansion model is non-radial.
Alternatively, if all outputs were expanded in the same proportion, the components of vector y need not be distinguished and equation (3) reduces to the model presented in ten Raa (2010).
The basic idea in (3) is that the demand for products is fulfilled by the industries producing them as primary outputs and secondary outputs are produced as by-products, i.e. negative inputs. The primary outputs of industries are maximized. It is a more flexible approach than ten Raa (2010) since the feasible set of equation (3) is larger, as demonstrated in Appendix 1.
The dual program equivalent to (3) is:
(4)
where the dual variables and solve (4) and match the shadow prices of the constraints of (3). Again, by the main theorem of linear programming, the primal and the dual program have equal solution values: .
ten Raa (2010) defined industrial organization efficiency of the industry k, as follows:
(5)
where is the ensemble efficiency determined by program (3), are the efficiency scores of each firm determined by the set of programs (1) and are the revenue shares of each firm evaluated at the prices determined by dual program (4)[3].
3. Industrial Specialization Efficiency
Ten Raa and Mohnen (2002, 2006) analyze the reallocation of factors between industries to decompose Total Productivity Growth. Ten Raa and Mohnen (2006) showed the interest of further decompose efficiency so as to consider the contribution of firms to it. The details are shown in the next section. With regard to the interpretation of efficiency measures, Shestalova (2002) further stated that the difference between augmented IOA and DEA lies on the interpretation of the frontier. The potential output is determined by the the best practices (DEA at industry level) or alternatively, by the reallocation of inefficiently allocated resources among industries (IOA in a multi-sectoral economy). To the best of our knowledge, the present paper is the first to simultaneously track the inefficiencies of the firms, the industries and the economy.
Industry efficiency is calculated with model (3) instead of a DEA-O CRS model. Then, we will work at the level of sectors , by pooling the vector of inputs and outputs within the firms of each industry k. The efficiency of the economy, , is obtained by
(6)
The equivalent dual program is:
(7)
where the dual variables w and p solve (7) and match the shadow prices of the constraints of (6). Again by the main theorem of linear programming: .
The underlying idea of (6-7) is to compute the efficiency when the maximum output is reached letting the reallocation of inputs among industries, not only within industry. Such maximum is the output that could be produced by the most efficient industries using the resources of non-efficient industries, i.e.: “how much textile could be produced using agriculture inputs” instead of “how much textile could be produced with the agriculture best-practice technique”, which is impossible. This is, somehow, a matter of opportunity cost and re-specialization of the output mix of the economy: the opportunity cost of producing a suboptimal output mix instead of the optimal one; this is, the cost in efficiency losses because of the wasting inputs in the production of inefficient commodities instead of in the most efficient ones (re-specialization of the output mix of the economy).
It is to be highlighted that in equations (6-7), the benchmarks are the best practices (firms) of the whole economy: The intensities in equation (6), ljh, are per firm and there is an activity constraint for each firm in the second set of constraints of equation (7). Intensities and activity constraints by industries would account for the best ‘industry-average’ practices, instead of the absolute best practices of the economy, not drawing the real production possibility frontier, but an average observed production. It is the same difference highlighted by ten Raa (2007), when discussing the difference between traditionally computed IO technical coefficients and technical coefficients obtained from best practices.
Analogous to (5), industrial specialization efficiency, , is:
(8)
where is the ensemble efficiency (whole economy efficiency) determined by program (6), are the efficiency scores of each industry determined by the set of programs (3) and are the revenue shares of each industry evaluated at the prices determined by dual program (7).
4. Efficiency of the Economy: Three way Decomposition
We are ready to present a single measure for the economy efficiency. Standard DEA techniques require a reference set and, therefore, are not applicable. Our measure, , will be derived internally. We build the efficiency measurement from the lowest level (firm) to the highest one (the whole economy) by a nesting decomposition of different efficiency measurements to isolate the effects at each level. Substituting (5) in (8) and reordering:
(9)
where is the industrial specialization efficiency calculated by (8), is the Organizational Efficiency of industry k, determined by (5), are the efficiency scores of each firm determined by the set of programs (1), and and are the revenue shares of each firm and each industry respectively, evaluated at the prices determined by dual programs (4) and (7).
At least theoretically the decomposition can be extended with an international/interregional level, bringing in the principle of comparative advantage, but this step requires comparable micro-data at an international level.
5. Application to the Andalusian Economy
Appendix 2 provides details about the database and computation and Appendix 3 shows the classification of industries/commodities. Table 1 summarizes the results of equations 1, 3 and 5: k is the industry code, ek is the industry k efficiency, is the organization efficiency of industry k and Hk is the firm’s efficiency weighed harmonic average of firms of industry k. #k is the number of firms within industry k.
The industries whose firms are technically inefficient could perform 1- Hk percentage points better by copying best – industry – practices. The industries whose firms may work better, ranging from 60% to 12% potential average improvement, are: Restaurants, bars and catering; Legal and Accounting services; Other services to firms; Wholesale trade; Advertising; Sale of motor vehicles and retail sale automotive fuel; Land Transport; Maintenance and repair of motor vehicles; Building completion; Architectural and engineering activities and related technical consultancy.
The industries whose organization is inefficient could perform Hk - ek percentage points better by exploiting economies or diseconomies of scope. Ranging from 79% to 36% of potential improvement, the industries with the worst organization are: Architectural and engineering activities and related technical consultancy; Real estate activities; Retail trade; Wholesale trade; Other services to firms; Supporting and auxiliary transport activities; Sale of motor vehicles and retail sale automotive fuel; Restaurants, bars and catering; Land transport; and Renting of machinery and equipment. Most of them are typically composed by small-sized firms. On the other hand, 29 industries[4] are fully efficient. Another 22 industries could improve as much as 10% of their performance by a better industrial organization.
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Table 1: Industry Efficiencies: Industry, Organizational, Firms mean.
k / ek / / Hk / #k / k / ek / / Hk / #k01 / 1 / 1 / 1 / 1 / 44 / 0.88 / 0.90 / 0.98 / 101
02 / 1 / 1 / 1 / 1 / 45 / 1 / 1 / 1 / 5
03 / 1 / 1 / 1 / 1 / 46 / 1 / 1 / 1 / 1
04 / 1 / 1 / 1 / 1 / 47 / 1.00 / 1.00 / 1.00 / 39
05 / 1 / 1 / 1 / 1 / 48 / 0.97 / 0.97 / 1.00 / 85
06 / 1 / 1 / 1 / 1 / 49 / 0.80 / 0.84 / 0.96 / 1574
07 / 1 / 1 / 1 / 8 / 50 / 0.57 / 0.66 / 0.87 / 1610
08 / 1 / 1 / 1 / 2 / 51 / 0.44 / 0.52 / 0.85 / 1468
09 / 0.81 / 0.81 / 1.00 / 135 / 52 / 0.68 / 0.78 / 0.86 / 946
10 / 0.92 / 0.92 / 1.00 / 167 / 53 / 0.16 / 0.21 / 0.75 / 5933
11 / 1.00 / 1.00 / 1 / 28 / 54 / 0.31 / 0.31 / 0.98 / 8887
12 / 0.99 / 0.99 / 1 / 41 / 55 / 0.69 / 0.75 / 0.93 / 673
13 / 0.99 / 0.99 / 1 / 43 / 56 / 0.00 / 0.00 / 0.40 / 2399
14 / 1.00 / 1.00 / 1.00 / 42 / 57 / 0.46 / 0.54 / 0.85 / 1995
15 / 1 / 1 / 1 / 4 / 58 / 1.00 / 1.00 / 1.00 / 19
16 / 1 / 1 / 1 / 5 / 59 / 0.53 / 0.55 / 0.97 / 966
17 / 0.80 / 0.81 / 0.99 / 559 / 60 / 0.97 / 0.98 / 1.00 / 417
18 / 0.89 / 0.89 / 1 / 82 / 61 / 1 / 1 / 1 / 1
19 / 1.00 / 1.00 / 1 / 22 / 62 / 1 / 1 / 1 / 1
20 / 0.92 / 0.92 / 1.00 / 113 / 63 / 0.75 / 0.80 / 0.94 / 332
21 / 0.72 / 0.75 / 0.96 / 200 / 64 / 0.28 / 0.28 / 1.00 / 808
22 / 0.90 / 0.91 / 0.99 / 117 / 65 / 0.59 / 0.62 / 0.95 / 480
23 / 0.81 / 0.82 / 0.98 / 254 / 66 / 0.57 / 0.62 / 0.92 / 292
24 / 0.97 / 0.97 / 1.00 / 53 / 67 / 0.92 / 0.93 / 0.99 / 65
25 / 0.90 / 0.91 / 0.99 / 202 / 68 / 0.21 / 0.39 / 0.55 / 1390
26 / 1 / 1 / 1 / 3 / 69 / 0.09 / 0.10 / 0.88 / 794
27 / 1.00 / 1.00 / 1 / 44 / 70 / 0.51 / 0.66 / 0.77 / 199
28 / 0.99 / 0.99 / 1.00 / 68 / 71 / 0.81 / 0.82 / 0.98 / 146
29 / 0.92 / 0.92 / 1.00 / 122 / 72 / 0.58 / 0.63 / 0.93 / 336
30 / 0.89 / 0.90 / 0.99 / 341 / 73 / 0.08 / 0.14 / 0.60 / 901
31 / 0.88 / 0.88 / 1.00 / 117 / 74 / 1 / 1 / 1 / 1
32 / 0.92 / 0.92 / 0.99 / 183 / 75 / 1 / 1 / 1 / 1
33 / 1.00 / 1.00 / 1.00 / 37 / 76 / 1 / 1 / 1 / 1
34 / 0.75 / 0.78 / 0.96 / 695 / 77 / 1 / 1 / 1 / 1
35 / 0.88 / 0.88 / 1.00 / 268 / 78 / 0.92 / 0.92 / 1.00 / 155
36 / 1.00 / 1.00 / 1 / 13 / 79 / 0.98 / 0.98 / 1 / 72
37 / 0.89 / 0.89 / 0.99 / 86 / 80 / 0.85 / 0.85 / 1.00 / 101
38 / 1 / 1 / 1 / 23 / 81 / 0.80 / 0.80 / 1.00 / 201
39 / 0.98 / 0.98 / 1.00 / 59 / 82 / 0.98 / 0.98 / 1.00 / 44
40 / 0.99 / 0.99 / 1.00 / 60 / 83 / 0.86 / 0.87 / 0.98 / 231
41 / 0.79 / 0.79 / 1.00 / 89 / 84 / 0.68 / 0.69 / 0.99 / 571
42 / 1 / 1 / 1 / 21 / 85 / 0.92 / 0.93 / 0.99 / 272
43 / 0.78 / 0.82 / 0.95 / 445 / 86 / 1 / 1 / 1 / 1
Key: ek: Efficiency of the industry k, eq. 3 eok: Organization Efficiency of the industry k, eq. 5